17edo tetrachords

日本語

Let a "17edo primary tetrachord" mean a set of four pitches in 17edo that span a perfect fourth (seven degrees) and include one of each of the following:

1. the unison - 0 (degrees of 17edo) - solfege name 'do'.
2. a second - includes 1 (ra, a minor second), 2 (ru, a neutral second), and 3 (re, a major second).
3. a third - includes 4 (me, a minor third), 5 (mu, a neutral third), and 6 (mi, a major third).
4. the perfect fourth - 7 (fa).

correspondance:

degrees	cents	name	solfege
0	0	unison	do
1	71	minor second (a.k.a third-tone)	ra
2	141	neutral second (a.k.a. two-thirds-tone)	ru
3	212	major second (a.k.a. tone)	re
4	282	minor third (a.k.a. subminor third)	me
5	353	neutral third	mu
6	424	major third (a.k.a. supermajor third)	mi
7	494	perfect fourth	fa

tetrachord notation

Tetrachord notation will show three scalar steps (as degrees of 17edo) separated by hyphens.

For instance, tetrachord 3-3-1 consists of
0 (do), the unison;
3 (re), a major second, 3 degrees up from 0;
6 (mi), a major third, 3 degrees up from 3; and
7 (fa), the perfect fourth, 1 degree up from 6.

The numbers in a tetrachord name will always add to 7.

17edo primary tetrachords

We have 9 primary tetrachords in 17edo.

tetrachord notation	solfege	name (suggestions?)	used in
1-3-3	do ra me fa	phrygian	diatonic (phrygian)
1-4-2	do ra mu fa
1-5-1	do ra mi fa	balkan, Hijaz (arabic)
2-2-3	do ru me fa	Bayyati (arabic)	17edo neutral scale (led)
2-3-2	do ru mu fa	Iraq (arabic)	17edo neutral scale (bish, fish, jwl)
2-4-1	do ru mi fa
3-1-3	do re me fa	aeolian	diatonic (aolian, dorian) ; scorp (mode 3)
3-2-2	do re mu fa	Rast (arabic)	17edo neutral scale (dril, gil, kleeth)
3-3-1	do re mi fa	ionian	diatonic (ionian, mixolydian)

Another way of showing them:

	ra	ru	re
me	1-3-3	2-2-3	3-1-3
mu	1-4-2	2-3-2	3-2-2
mi	1-5-1	2-4-1	3-3-1

17edo tetrachords complete

A more generalized tetrachord system would allow multiple seconds or multiple thirds: for instance, 1-1-5 or 5-1-1. Thus, a complete chart of 17edo tetrachords looks like this (with primary tetrachords in bold):

1-1-5	2-1-4	3-1-3	4-1-2	5-1-1
1-2-4	2-2-3	3-2-2	4-2-1
1-3-3	2-3-2	3-3-1
1-4-2	2-4-1
1-5-1

Thus, by allowing multiples seconds or multiple thirds, we add 6 new tetrachords to our 9 primary tetrachords, for a total of 15. Our new ones:

tetrachord notation	solfege	name (suggestions?)	used in
1-1-5	do ra ru fa
1-2-4	do ra re fa
2-1-4	do ru re fa
4-1-2	do me mu fa
4-2-1	do me mi fa
5-1-1	do mu mi fa

See also: tetrachord, 22edo tetrachords, Tricesimoprimal Tetrachordal Tesseract.